Moving only one vertex of the edge
You can use AltV for Rip fill.



Normally you could just press G with the vertex selected (or GG if you wanted to slide a vertex along the edge).
However, since the vertex you want to move has other edges connected to it, you must create a new, separate vertex with the one edge attached (Rip, V), and then fill in the newly formed hole (Fill, F). The Rip Fill tool does these two operations automatically.

I assumed that there is only one vertex at the corner. If you have multiple vertices in the same place (doubles), then you need only to press G with the right one selected.